Lyapunov-Schmidt 约化有效性域的显式二阶界
Explicit Second-Order Bounds on the Domain of Validity for Lyapunov-Schmidt Reduction
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中文总结 AI 辅助
本文针对 Lyapunov-Schmidt 约化，利用向量场 Hessian 的二阶条件推导有效性域的显式界，并在类 Hopfield 网络及正则图叉形分岔中验证其适用性。
中文摘要 AI 辅助
Lyapunov-Schmidt 约化是一种在高维系统分岔分析中广泛使用的降维技术。经典表述虽然保证了约化方程局部存在性，但通常缺乏关于这些表示能够忠实捕捉完整分岔结构的邻域大小的显式定量估计。在近期工作中，我们通过利用向量场的一阶条件以及隐函数定理的定量结果，推导了该约化有效性域的界，从而解决了这一局限。本文进一步深入，采用包含向量场 Hessian 的二阶条件来发展这些界，并获得了一组新结果。随后，我们研究了这些新推导的界在类 Hopfield 网络动力系统中的适用性，在连通正则图上评估了这些界在叉形分岔中的表现，并最终考察了该类系统中两个认证域之间的关系。
英文摘要
Lyapunov-Schmidt reduction is a widely used dimensionality reduction technique for bifurcation analysis in high-dimensional systems. While classical formulations guarantee the local existence of reduced-order equations, they typically lack explicit quantitative estimates on the size of the neighbourhoods in which these representations faithfully capture the full bifurcation structure. In recent work, we have addressed this limitation by deriving bounds on the domain of validity of this reduction using first-order conditions on the vector field together with a quantitative result for the implicit function theorem. In this article we explore beyond, and adopt second-order conditions that incorporate the Hessians of the vector field to develop these bounds, and obtain a new set of results. We then investigate the applicability of these newly derived bounds to Hopfield-like networked dynamical systems, evaluate these bounds for pitchfork bifurcations on connected regular graphs, and finally examine the relationship between the two certified domains for this class of systems.
发表机构
- Indian Institute of Technology Bombay（印度理工学院孟买分校）
- Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。
